{"paper":{"title":"Empires and Percolation: Stochastic Merging of Adjacent Regions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"D. J. Aldous, J. R. Ong, W. Zhou","submitted_at":"2009-11-03T15:57:16Z","abstract_excerpt":"We introduce a stochastic model in which adjacent planar regions $A, B$ merge stochastically at some rate $\\lambda(A,B)$, and observe analogies with the well-studied topics of mean-field coagulation and of bond percolation. Do infinite regions appear in finite time? We give a simple condition on $\\lambda$ for this {\\em hegemony} property to hold, and another simple condition for it to not hold, but there is a large gap between these conditions, which includes the case $\\lambda(A,B) \\equiv 1$. For this case, a non-rigorous analytic argument and simulations suggest hegemony."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.0601","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}